There are two closely related constructions, due Yuri Manin, for finitely generated quadratic algebras, and due Tambara, for finite dimensional algebras.
Tambara’s universal coacting bialgebra
If is a finite dimensional (associative unital) -algebra, and the functor where is a -algebras has a left adjoint .
where are arbitrary -algebras. has a canonical structure of a coalgebra, making it into a -bialgebra, the universal coacting bialgebra.
Daisuke Tambara, The coendomorphism bialgebra of an algebra, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 37, 425-456, 1990 pdf
Tambara’s construction is dual to the universal measuring coalgebra of Sweedler.
Manin’s universal coacting bialgebra
In a similar way to above, one utilizes the adjunction between inner hom and functor for quadratic algebras.
Yu. I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988.